202 Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209

Frontiers of Information Technology & Electronic Engineering

www.jzus.zju.edu.cn; engineering.cae.cn; www.springerlink.com

ISSN 2095-9184 (print); ISSN 2095-9230 (online)

E-mail: jzus@zju.edu.cn

ApplyingRational Envelope curves for skinning purposes∗

Kinga KRUPPA1,2

1Faculty of Informatics, University of Debrecen, Debrecen H-4028, Hungary2Doctoral School of Informatics, University of Debrecen, Debrecen H-4028, Hungary

E-mail: kruppa.kinga@inf.unideb.hu

Received July 25, 2019; Revision accepted June 27, 2020; Crosschecked Sept. 16, 2020; Published online Nov. 28, 2020

Abstract: Special curves in the Minkowski space such as Minkowski Pythagorean hodograph curves play animportant role in computer-aided geometric design, and their usages are thoroughly studied in recent years. Bizzarriet al. (2016) introduced the class of Rational Envelope (RE) curves, and an interpolation method for G1 Hermitedata was presented, where the resulting RE curve yielded a rational boundary for the represented domain. We nowpropose a new application area for RE curves: skinning of a discrete set of input circles. We show that if we donot choose the Hermite data correctly for interpolation, then the resulting RE curves are not suitable for skinning.We introduce a novel approach so that the obtained envelope curves touch each circle at previously defined pointsof contact. Thus, we overcome those problematic scenarios in which the location of touching points would not beappropriate for skinning purposes. A significant advantage of our proposed method lies in the efficiency of trimmingoffsets of boundaries, which is highly beneficial in computer numerical control machining.

Key words: Medial axis transform; Envelope; Interpolation; Skinning; Circlehttps://doi.org/10.1631/FITEE.1900377 CLC number: TP391

1 Introduction

Medial axis transform (MAT) has been thor-oughly studied in both computer graphics and imageprocessing. For a given planar domain, the medialaxis (MA) is the locus of centers of maximal inscribeddisks, and the MAT can be obtained by lifting theMA to the space using the radii of the inscribeddisks. Once we obtain the MAT, we can reconstructthe boundary of the domain using the well-knownenvelope formula introduced by Choi et al. (1997,1999). Curves in the Minkowski space are suitablefor describing MATs; however, only some specialtypes of curves describe domains whose boundariesare rational. Moon (1999) showed that Minkowski

* This work was supported by the construction EFOP-3.6.3-VEKOP-16-2017-00002. The project was supported by the Eu-ropean Union, co-financed by the European Social Fund. Openaccess funding was provided by University of Debrecen

ORCID: Kinga KRUPPA, https://orcid.org/0000-0001-5359-2829c© The Author(s) 2020

Pythagorean hodograph (MPH) curves are such, andtheir yielded envelopes are Pythagorean hodographcurves; thus, the envelopes and their offsets are ra-tional as well.

Nowadays, several Hermite interpolation meth-ods have been proposed using MPH curves so thatthe resulting envelope is rational (Kim and Ahn,2003; Kosinka and Jüttler, 2006, 2009; Kosinka andŠír, 2010; Kosinka and Lávička, 2011; Bizzarri et al.,2019). Bizzarri et al. (2016) showed that a broaderclass of curves exists in R

2,1, which yields rationalboundaries: the so-called Rational Envelope (RE)curves. Thus, if only the rationality of the envelopeis required, one can rely on RE curves, which areeasy to compute, whereas one has to restrict oneselfto MPH curves only if the rationality of offsets isalso needed. The authors described a G1 interpola-tion method to construct an RE interpolant that canbe used to create rational blending surfaces betweencanal surfaces.

Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209 203

Now we introduce a quite different area incomputer-aided geometric design (CAGD) calledskinning. Standard point-based curve and surfacemodeling has always played an important role inCAGD, but there has been a growing interest inmodeling based on different objects such as circlesand spheres. Skinning has become known as themethod of constructing a pair of at least G1 con-tinuous splines for a predefined sequence of circlesthat touches each circle at one point. The idea canalso be extended to three-dimensional (3D) model-ing, leading to the skinning of an input set of spheres.In recent years, several skinning methods have beendeveloped by Kunkli and Hoffmann (2010), Banaet al. (2014), Bastl et al. (2015), and Kruppa et al.(2019). In addition, skinning can be applied in vari-ous fields like computer animation (e.g., ZSpheres R©

(Pixologic Inc., 2020) and SporeTM (Electronic ArtsInc., 2008)), molecular biology, and medical imageprocessing (Rossignac et al., 2007; Slabaugh et al.,2008, 2010; Piskin et al., 2017).

Skinning and boundary reconstruction are sim-ilar; however, their problem settings are fundamen-tally different. In skinning, the reconstruction of aboundary is not possible because there is no initialdomain assumed at all. Despite this fact, on the ba-sis of the similarities between the two, we propose toapproach skinning from the aspect of MAT and useyielded envelope curves.

2 Motivation and related study

Let us first consider a parametric curve y(t) =

(y(t), r(t)) as the MAT of a domain. Then theboundary (envelope) of the domain can be recon-structed using the envelope formula (Choi et al.,1997, 1999):

x±(t)

=y(t)− r(t)r′(t) y′(t)± y′⊥(t)

√‖y′(t)‖2 − r′(t)2

‖y′(t)‖2.

(1)

Next, let us overview the G1 Hermite interpo-lation method proposed by Bizzarri et al. (2016) be-cause the RE curve construction method will be usedfor our new skinning approach. In Section 2.2, we ex-amine the standard skinning method of Kunkli andHoffmann (2010).

2.1 G1 Hermite interpolation method yield-ing rational envelopes

Given two points and two vectors as the inputHermite data for interpolation, let P0 and P1 denotethe endpoints, and t0 and t1 tangent vectors. Markprojection as

�n = (nx, ny) if n = (nx, ny, nz) and

n ∈ R2,1. Mark rotation as n⊥ = (ny,−nx) if n =

(nx, ny) and n ∈ R2.

Directly applying the envelope formula(Eq. (1)), we can obtain the touching points Q±

i

(i ∈ {0, 1}) of the corresponding envelope curve asfollows:

Q±i =

�Pi − Piz

tiz�ti ±

�ti

⊥√‖�ti‖2 − t2iz

‖�ti‖2

. (2)

The construction relies on only one branch, andwe define the touching point Qi as Q+

i . We thendefine appropriate tangent vectors:

vi = αi(Qi −

�Pi)

⊥

‖Qi −�Pi‖

, αi ∈ R. (3)

Once we obtain Qi and vi, construct the planarcurve x(t) (t ∈ [0, 1]) that interpolates Qi and vi,for example, by Hermite interpolation.

The final step is to construct the MA y(t) as aone-sided offset of x(t) with varying distance:

y(t) = x(t) + r(t)x′⊥(t)‖x′(t)‖ = x(t) + f(t) x′⊥(t),

which is rational only if f(t) = r(t)/‖x′(t)‖ isa rational function. To assure that the resultingcurve interpolates the initial input data Pi and ti,f(t) : [0, 1] → R must be constructed as a polyno-mial function with the following boundary conditions(i ∈ {0, 1}):

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

f(i) =Piz

‖vi‖,

f ′(i) = −�ti · (f(i) x′′(i)− v⊥

i )�ti · vi

.(4)

By lifting the MA back to the space, the final formof the interpolant is as follows:

y(t) =

[xx(t) + x′

y(t)f(t), xy(t)− x′x(t)f(t),

f(t)√

x′2x (t) + x′2

y (t)

]. (5)

204 Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209

The resulting y(t) is a so-called RE curve, a curveyielding a rational envelope. For the construction,refer to Fig. 1. Bizzarri et al. (2016) proved thatthe envelope is indeed rational. Moreover, if x(t) isa Pythagorean hodograph (PH) curve, then y(t) isan MPH curve, and the offsets of the envelope arerational as well.

P0t0

P1t1

Q0

v0Q1

v1x(t)

y(t)

y(t)–

Fig. 1 Construction of the Rational Envelope (RE)curve y(t), where G1 interpolates the input data Pi

and ti (i ∈ {0, 1}) using the method of Bizzarri et al.(2016)

2.2 Skinning of circles

As mentioned in Section 1, skinning is a tech-nique used in CAGD for modeling. We introducethe skinning method of Kunkli and Hoffmann (2010).Given an ordered set of circles, skinning is the con-struction of two G1 continuous curves touching eachof the given circles at one point separately. Oneof the most important steps is the localization ofthe touching points (Fig. 2). The touching pointsfor the first and last circles are determined by thetwo common outer tangents of the circles; for theother circles, circle triplets are considered. For eachtriplet, we find two special solutions to the Apollo-nius problem, using which we can define the touch-ing points for the middle circle. Once we determinepoints of contact, they are separated into two groupsfor “left” and “right” skins. Finally, Hermite inter-polation curves are constructed between every twoneighboring circles for which the tangent lengths arecomputed using the radical lines of the circle pairs.The resulting G1 continuous splines are called skinsof the circle set.

The algorithm in Kunkli and Hoffmann (2010)generally provides good results for various input sets,and the skins respond dynamically when the usermodifies the positions or the radii of the circles dur-ing modeling. Because the touching points are de-rived using specific solutions to the Apollonius prob-

lem, it is also guaranteed that they never lie insideany of the neighboring circles (Fig. 3).

3 Applying Minkowski Pythagoreanhodograph/Rational Envelope curvesfor skinning purposes

In this section, we propose a new applicationarea for MPH/RE curves: skinning discrete sets ofcircles. As the problem setting of skinning involvesfinding bounding curves for an admissible set of pre-defined circles, we can define the appropriate inputset as mentioned in Kunkli and Hoffmann (2010).Definition 1 Given the ordered set of circlesC = {c1, c2, . . . , cn} (n ∈ N) and the correspond-ing disks D = {d1, d2, . . . , dn}, C is an admissibleconfiguration for skinning if it satisfies the followingconditions (Fig. 4):

1. di �⊂n⋃

j=1,j �=i

dj , i ∈ {1, 2, . . . , n};

2. di ∩ dj = ∅, i, j ∈ {1, 2, . . . , n} , j /∈

ci−1

Wi

ci

ci+1

Wi+1

Wi−1Wi

Wi−1

Wi+1

+

+

−

−

+

−

Fig. 2 Determining touching points by the methodof Kunkli and Hoffmann (2010) for a sequence of cir-cles: for each circle ci, the touching points are deter-mined using specific solutions to the Apollonius prob-lem (red and green); in the case of the first and lastcircles, the common outer tangents are used. Refer-ences to color refer to the online version of this figure

Fig. 3 Skinning circles with the method given byKunkli and Hoffmann (2010)

Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209 205

{i− 2, i− 1, i, i+ 1, i+ 2};3. di−1 ∩ di+1 �= ∅ =⇒ di−1 ∩ di+1 ⊂ di,

i ∈ {2, 3, . . . , n− 1}.To attain an envelope for the admissible se-

quence of circles C, we need to construct inputdata for the interpolation. Let us use cyclographicmapping so that we can map the circles to pointsin R

2,1. Thus, we obtain the sequence of pointsP = {P1,P2, . . . ,Pn}, where for each circle ci withcenter Oi(Oix , Oiy ) and radius ri, the correspondingpoint Pi ∈ R

2,1 is as follows:

Pi = (Oix , Oiy , ri). (6)

The next step is to define T = {t1, t2, . . . , tn},the sequence of tangent vectors. However, becausewe do not have any additional information about thecircles, there are no initial tangent vectors provided.We may define the vectors using, e.g., the Catmull-Rom spline interpolation, so that each ti ∈ R

2,1

tangent vector is defined as ti = λ · (Pi+1 − Pi−1)

(i ∈ [2, n−1], λ ∈ R\ {0}). For the first and last cir-cles, the vectors are defined as t1 = λ · (P2−P1) andtn = λ · (Pn−Pn−1). Furthermore, using Eq. (2) wecalculate positions of the touching points, and afterconstructing y (the RE curve), we obtain the enve-lope. Generally, this provides adequate output; how-ever, in the case of intersecting circles, the resultingtouching point can get inside the neighboring circle,and the envelope can cut into the circles (see an ex-ample in Fig. 5). In addition, for MPH/RE curves,tangent vectors must be space-like (i.e., their anglewith the x-y plane must be less than 45◦). Taking allthese factors into consideration, we cannot use REcurves for skinning by arbitrarily choosing tangentvectors.

To solve this problem, we can approach it froman inverse aspect. In Section 2, we have discussedthat the method of Kunkli and Hoffmann (2010) lo-calizes the touching points by solving the Apolloniusproblem; thus, it guarantees that the points never lieinside any of the neighboring circles. Our idea is tolocalize the touching points at first and reconstructprecisely that tangent vector with which we wouldobtain these touching points.

3.1 Reconstructing tangent vectors

For each circle ci in C, fix two points that wewant to use later as the touching points of the enve-lope. Mark the left point as W+

i and right one as

c2

c3 c4c5 c6

c1

Fig. 4 An example of nonadmissible configuration:by changing, e.g., the positions of the red circles ac-cording to Definition 1, we can define both “left” and“right” touching points on every circle. References tocolor refer to the online version of this figure

Fig. 5 Using seemingly appropriate tangent vectorsthat are chosen freely, we observe that the touchingpoints may lie inside the circles

W−i . The aim is to determine the tangent vector for

which we obtain W±i = Q±

i using Eq. (2).It is known that the envelope formula describes

a geometric construction (Pottmann and Peternell,1998; Peternell et al., 2008). Kunkli (2009) also pre-sented different ways to obtain touching points forskinning, one of which was using the geometrical ap-proach to create the corresponding touching points.We use this approach as an inspiration to reconstructthe tangent vector ti from the touching points.

Let W+i and W−

i denote the left and righttouching points on ci, respectively (Fig. 2). We aimto construct Si, the endpoint of the desired tangentvector. Define line ei =

←−−−−→W+

i W−i and line fi so that

�Pi ∈ fi and fi ⊥ ei. Let Ii denote the intersectionpoint of ei and fi. Si can now be defined as theinverse point of Ii with respect to circle ci.

Six = Pix + r2iW+

iy−W−

iy

γi, (7)

Siy = Piy − r2iW+

ix−W−

ix

γi, (8)

γi =Pix

(W−

iy−W+

iy

)+ Piy

(W+

ix−W−

ix

)

+W+iyW−

ix−W+

ixW−

iy. (9)

Once the endpoint Si is obtained, the tangent

206 Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209

vector ti ∈ R2,1 can be practically defined as follows:

ti =

[r2iγi

(W+

iy−W−

iy

),r2iγi

(W−

ix−W+

ix

), −ri

].

(10)

Geometric construction of ti is shown in Fig. 6.Because our construction is exactly the inverse ofthe standard geometric construction of the envelopepoints, it is trivial that once we define ti this way,the points Q+

i and Q−i obtained using Eq. (2) are

the same as W+i and W−

i , respectively. Besides,choosing the tangent vector this way assures that tiis always space-like.

Pi

fi

ei

Ii

Wi−

Wi+

vi+

Si

ti

Fig. 6 Reconstructing tangent vector ti using fixedtouching points W

+i and W

−i . The construction is

based on the geometric properties of the envelope ofa family of circles

3.2 Constructing envelope curves

We have seen that the RE method defined byBizzarri et al. (2016) uses the αi values in Eq. (3) asfree parameters, which essentially affect the shape ofthe resulting envelope, as y(t) is constructed fromthe planar curve x(t). Because we now use thetouching points mentioned in Kunkli and Hoffmann(2010), it is natural to choose the tangent length thatthe authors suggest. Let li denote the radical line ofci and ci+1. To assure a more aesthetically satisfy-ing result, we differ the starting and ending tangentlengths for each neighboring circle pair. Thus, forthe circle pair ci and ci+1, the corresponding tan-gent vectors vi and vi+1 for the touching point aredefined as follows:

vj = αj

(W+j −

�Pj)

⊥

‖W+j −

�Pj‖

, j ∈ {i, i+ 1}. (11)

The αj values can be determined using the Euclideandistance between W+

j and li:

αj = 2 dist(W+j , li). (12)

To define the MA, we now need only to follow theremaining steps of the algorithm in Bizzarri et al.(2016). After defining x(t) as a rational planar curveinterpolatingW+

i , W+i+1,vi, and vi+1 (e.g., as a Her-

mite arc), we construct the polynomial function f

with the boundary conditions (Eq. (4)), and obtainthe final RE curve in the form of Eq. (5). The result-ing y is an RE curve, for which by applying Eq. (1),the resulting envelope curves are rational.

Fig. 7 shows our solution so that the resultingenvelope curves are always adequate for skinning.Bizzarri et al. (2016) proved that if we constructx(t) as a PH curve, not only the envelope curves butalso their offsets are rational. Fig. 8 shows such ascenario in which the resulting offsets are rationaltoo.

4 Discussion

By applying our new algorithm for more com-plex systems of circles, we can create aesthetic

Fig. 7 For the same input as in Fig. 5, we observe thatthe touching points now lie outside the neighboringcircles; thus, if we reconstruct the tangent vectorsfrom the previously chosen touching points, we areable to use the RE curve for skinning

Fig. 8 Minkowski Pythagorean hodograph curve usedfor skinning with our new approach. Not only theenvelope curves but also the offsets are rational

Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209 207

models. In Fig. 9, we can see various examples: aseahorse (which has been used as a model in ex-isting skinning methods, motivated by Kunkli andHoffmann (2010)), a snake, and a pair of knife andspoon. We give adequate results for shapes that in-clude many more circles with different radii as well.Because the concept is based on the MAT and thusthe input circles behave as maximal inscribed disksof the planar shape, the resulting envelope may havesingularities in extreme positions. However, we cansimply overcome these situations by adding more cir-cles to describe the desired shape more precisely.

(a) (b) (c)

Fig. 9 Complex and aesthetic two-dimensional mod-els created with our proposed method using the en-velopes of RE curves: (a) seahorse; (b) snake; (c) apair of knife and spoon

The most significant advantage of applying REand MPH curves for skinning is that they can be usedefficiently in computer numerical control (CNC) ma-chining. The detection of self-intersections of the off-set curves and the trimming process are fundamen-tal issues, especially in CNC machining and robotpath planning, as the intersection of the offsets wouldcause machine failure in the milling process. Offsetcurves are trimmed to remove unwanted parts basedon their self-intersections, which is a highly expen-sive and computationally difficult operation for gen-eral free-form curves. However, using MATs, we canconstruct them in an easy and straightforward way(Choi et al., 1999, 2008; Cao and Liu, 2008). For agiven y(t) (RE/MPH curve), the inner δ-offsets ofthe envelope curves can be directly constructed as

x±δ (t) = y(t) + (r(t) − δ)m±(t), (13)

where m±(t) is obtained by orthogonally projectingthe normal vectors of the y curve. Trimming the

δ-offset curves has now become quite simple: thosex±δ (t0) parts have to be trimmed where r(t0) < δ.

Thus, if we create circle-based models (such as logos,ornaments, and artistic shapes) by skinning themwith MPH/RE curves, we assure that CNC machinescan efficiently mill them as we can provide trimmedoffsets. An example of the trimming process can beseen in Fig. 10.

(a) (b)

Fig. 10 Envelope curves with untrimmed (a) andtrimmed (b) offsets: using RE curves for skinningpurposes, the offset trimming process is efficient andstraightforward

5 Conclusions

The RE curves were introduced by Bizzarriet al. (2016), and they presented a G1 interpolationmethod to construct an RE curve in the R

2,1

Minkowski space, which yields a rational envelope.Moreover, they can be constructed so that the offsetsare rational as well. In this study, we propose a newapplication for RE/MPH curves: skinning a discreteset of circles. To obtain the envelope, we firstneed to define the appropriate Hermite input data.We show that circles can be regarded as spatialpoints using cyclographic mapping, but there is noinformation about the tangent vectors that shouldbe used for the interpolation. Unquestionably, theshape of the envelope fundamentally depends onthe chosen tangent vectors. Even by choosing aseemingly suitable method to define these vectors,the resulting touching points and shape of theenvelope are sometimes problematic. To resolvethis problem, we offer an inverted approach: wepredefine the positions of the touching points (by the

208 Kruppa / Front Inform Technol Electron Eng 2021 22(2):202-209

method of Kunkli and Hoffmann (2010)), and thencalculate which is the appropriate tangent vectorthat would precisely provide the same touchingpoints. We also give an exact calculation to fix thefree parameters of the method by Bizzarri et al.(2016), which significantly affect the shape of theenvelope as well. Thus, our method does not needa numerical optimization afterward to generatevalid output. We give an exact solution, and thus,guarantee that there exists only one envelope fora user-defined circle sequence. In the study, wegive all the necessary algorithmic details for theproper construction of the RE curves, for whichthe envelope curves are adequate for skinning.We also provide several examples of our proposedconstruction. A significant advantage of this newapproach is the efficient offset trimming process,which is fundamentally beneficial, for example, inCNC machining. For future study, we plan to extendour algorithm to 3D modeling to provide anotherapplication for RE curves: skinning a discrete set ofspheres.

Compliance with ethics guidelinesKinga KRUPPA declares that she has no conflict of

interest.

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